Modern signal, image, and information processing are relying more and more on the use of matrix albebra as a basic computational tool. In addition, many problems require a real-time computational capability. For these reasons, emphasis is currently being placed on the development of dedicated electronic parallel-processing devices using VLSI/VHSIC technology for performing the intensive computations required in numerical matrix algebra. Recent interest of using optical techniques for performing matrix operations is becoming more clearly evident. The innate parallelism, non-interfering communication and wide bandwidth of optical processing systems have successfully demonstrated its strengths in performing convolutions and correlations as well as a variety of linear transform operations. Optical or photonic computing will have significant impact on general matrix computation, provided that new concepts for improving the precision are developed.
One development which has improved the precision of optical computations employs an algorithm for performing multiplications and additions using optical convolving devices. This was the subject matter of a presentation by H. J. Whitehouse et al, entitled "Aspects of Signal Processing with Emphasis on Underwater Acoustics", Part 2, G. Tacconi, edition, Reidel, Dordrecht, The Netherlands (1977) and D. Psaltis et al, in their article entitled "Accurate Numerical Computation By Optical Convolution", in 1980 International Optical Computing Conference Book II. W. T. Rhodes edition, Proc., SPIE 232, 151 (1980). The algorithm identified in these articles has been become popularly known as the digital multiplication by analog convolution (DMAC) algorithm. For the case of radix 2, for example, the DMAC algorithm is novel in that binary numbers may be added without carries if the output is allowed to be represented in a mixed binary format. In the mixed binary format, as in binary arithmetic, each digit is weighted by a power of 2, but unlike binary arithmetic, the value of each digit can be greater than 2. It is the elimination of the need for carries that makes this technique particularly attractive in terms of optical implementation. The DMAC algorithm has been used in optical architectures for performing matrix multiplication involving numbers expressed in fixed-point form. Such applications are documented for example, by the article by W. C. Collins et al entitled "Improved Accuracy for an Optical Iterative Processor," in Bragg Signal Processing and Output Devices, B. V. Markevitch and T. Kooij, editors, Proc., SPIE 352, 59, (1983); R. P. Bocker et al in their article entitled, "Electrooptical Matrix Multiplication Using the Twos Complement Arithmetic for Improved Accuracy," Appl. Opt. 22, 2019 (1983); by P. S. Guilfoyle, in his article entitled "Systolic Acousto-optic Binary Convolver," Opt. Engr. (23)/(1), 20 (1984), and A. P. Goutzoulis, in his article entitled "Systolic Time-Integrating Acoustooptic Binary Processor," Appl. Opt. 23, 4095 (1984). A floating-point form of the DMAC algorithm has been expressed in an optical architecture by articles authored by R. P. Bocker et al, see "Optical Matrix-Vector Multiplication Using Floating-Point Arithmetic," in Optical Computing Technical Digest, TuD3-1, OSA (1985), and "Optical Fixed-Point Arithmetic," presented at SPIE Conference No. 564, Real Time Signal Processing VIII, W. J. Miceli and K. Bromley, chairmen, San Diego (August 1985).
With respect to the implementation of the DMAC algorithm however, as expressed above, there continues to be one main criticism when it is used in matrix multiplication. This criticism is the need for high-speed electronic analog-to-digital converters, required for converting mixed binary back to pure binary. In addition, the DMAC algorithm does not take full advantage of the intrinsic parallelism of optics.
A second optical technique for improving precision is based on the use of the residue number system. The literature is replete with references to this system, among which is the article by, A. Huang, entitled "The Implementation of a Residue Arithmetic Unit Via Optical and Other Physical Phenomena," Digest of the International Optical Computing Conference, IEEE Catalog 75-CH0941-5C, (New York 1975), and A. Huang et al, article entitled "Optical Computation Using Residue Arithmetic," Applied Optics 18, 149 (1979), and F. A. Horrigan et al, entitled "Residue-Based Optical Processor," in Optical Processing Systems, W. T. Rhodes, editor, Proc., SPIE 185, 19 (1979), to name a few. The residue number system offers the appealing feature of carry-free addition, subtraction, and multiplication, thus making it very attractive for parallel processing. However, the main disadvantages of the residue number system concern the awkwardness of conversion from a residue form to a standard number representation, the difficulty in performing division, and the complexity in determining the correct sign of a subtraction operation.
The modified signed-digit (MSD) number representation has been set forth in the literature as a third approach for improving the precision of optically performed computations. The article by A. Avizienis, entitled "Signed-Digit Number Representations for Fast Parallel Arithmetic," IRE Trans. Electronic Computers EC-10, 389 (1961), describes a class of number representations otherwise known as signed-digit representations. This number system offers carry-free addition and subtraction, fixed-point as well as floating-point capability, and the potential for performing divisions optically. However, the paper, while presenting certain logic design problems, did not provide the necessary optical architecture for actually performing such computations.
Thus, there exist in the state-of-the-art the necessary optical architectures to implement the modified signed-digit number representation in addition, subtraction and multiplication, so that fully parallel carry-free operation is provided for with reduced complexity to realize the advantages of optical signal processing.